By Simpson D.J.W.
Real-world platforms that contain a few non-smooth swap are frequently well-modeled through piecewise-smooth platforms. even though there nonetheless stay many gaps within the mathematical concept of such platforms. This doctoral thesis offers new effects relating to bifurcations of piecewise-smooth, non-stop, self reliant platforms of normal differential equations and maps. a variety of codimension-two, discontinuity triggered bifurcations are opened up in a rigorous demeanour. a number of of those unfoldings are utilized to a mathematical version of the expansion of Saccharomyces cerevisiae (a universal yeast). the character of resonance close to border-collision bifurcations is defined; particularly, the curious geometry of resonance tongues in piecewise-smooth non-stop maps is defined intimately. Neimark-Sacker-like border-collision bifurcations are either numerically and theoretically investigated. A finished history part is comfortably supplied for people with very little event in piecewise-smooth structures.
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Extra resources for Bifurcations in piecewise-smooth continuous systems
2). First, z ∗(R) may be an attracting focus. This case is studied in Sec. 2. 1). 2) has two real-valued eigenvalues, λ1,2 (λ1 ≥ λ2 ). 1) has no nonlinear terms, the parts of the associated eigenvectors, v1,2 , that lie in the right half-plane form invariant, semiinfinite lines, γ1,2 . When z ∗(R) is admissible, each γi intersects z ∗(R) in the right half-plane, otherwise the straight line extension of each γi intersects the virtual equilibrium, z ∗(R) , in the left half-plane. 12) . 13) − δR , hence γ1,2 c1,2 = y ∗(R) − m1,2 x∗(R) = µ δR =− −τR + µλ1,2 .
Bifurcations November 26, 2009 15:34 World Scientific Book - 9in x 6in Chapter 2 Discontinuous Bifurcations in Planar Systems As detailed in Sec. 5, a bifurcation resulting from the collision of an equilibrium with a switching manifold in a piecewise-smooth, continuous ODE system is known as a discontinuous bifurcation. Discontinuous bifurcations in systems of dimension three or greater may generate complicated invariant sets, such as homoclinic orbits satisfying Sil’nikov’s condition [Llibre et al.
It is conjectured that if n ˆ = 0 then no bifurcation occurs at µ = 0. This has been verified for many examples but to the author’s knowledge has yet to be proved [Leine (2006); Leine and Nijmeijer (2004)]. If n ˆ = 1, the discontinuous bifurcation is called a single-crossing bifurcation. Leine et al. observe that single-crossing bifurcations invariably appear to be the analogue of some familiar smooth bifurcation. Fig. 5 illustrates discontinuous analogues of saddle-node and Hopf bifurcations (see [Leine and Nijmeijer (2004)] for transcritical and pitchfork-like bifurcations).